1. Field of the Invention
The present relates generally to a method and system for predicting a future state of a vehicle and, more particularly, to such a system utilizing the Unscented Transform (UT) and Numerical Integration (NI) together to stochastically predict the future state.
2. Description of the Related Art
Various vehicular collision detection approaches exist. Some examples include intelligent parking assist and pre-crash sensing. Production systems use radars, laser radars, cameras, or ultrasonic sensors to detect obstacles. However, the majority of these systems do not provide an assessment of another vehicle's future path of travel, but instead solely rely on the proximity of the vehicle as sensed by the sensor.
In order to predict the future position of a vehicle it is conventional to utilize a mathematical nonlinear model of the vehicle. Conventional models include the constant acceleration kinematic model (CA), the kinematic unicycle model (KU), the kinematic bicycle model (KB) or the classic bicycle model (CB). Each model consists of differential equations which, when solved, represent the dynamic action of the automotive vehicle.
Once the model has been selected, one previously utilized approach was to utilize Kalman Prediction to predict the future horizon position of the vehicle at time To+Th where Th equals the horizon time offset into the future from the current time To. Since all of the models are nonlinear, continuous time models, in order to apply the discrete Kalman equations, the nonlinear continuous time models must first be linearized through derivation of the Jacobian state transition matrix, ∇F, and the input gain matrix, ∇G. In addition, Kalman Prediction requires that a discrete time system model propagate forward through the prediction horizon Th. Therefore, at each propagation step, Tstep, the linearized, continuous-time system must be discretized as follows:
                                          x            ⁡                          (              t              )                                ~                      ∇                          Fx              ⁡                              (                t                )                                                    +                  ∇                      Gu            ⁡                          (              t              )                                                      ⇓                                    x          ⁡                      [                          k              +              1                        ]                          =                                            A              d                        ⁢                          x              ⁡                              [                k                ]                                              +                                    B              d                        ⁢                          u              ⁡                              [                k                ]                                                        where x(t) is the continuous state, x[ ] is the discretized state, Ad is an n×n matrix, Bd is an n×p matrix, n is the number of states, p is the number of inputs, and Ad and Bd are the discretized system using the sample time Tstep.
While Kalman Prediction has proven sufficiently accurate in automotive systems for predicting the future position of the vehicle, Kalman Prediction is necessarily computationally intensive. Since microprocessors of the type used in automotive vehicles, for cost considerations, are not fast relative to personal computers, the computational-intensive equations required by Kalman Prediction mandate relatively long time steps Tstep between sequential equations. This, in turn, can introduce error into the predicted future position of the vehicle.
The UT is a method for calculating the statistics of a random variable which undergoes a nonlinear transformation. The intuition behind the UT is that it is easier to approximate a Gaussian distribution than it is to approximate an arbitrary nonlinear function or transformation. In contrast, the Extended Kalman Filter (EKF) approximates a nonlinear function using linearization, and this can be insufficient when the selected model has large nonlinearities over short time periods.
The UT has been applied to the Kalman-filtering problem to form the well-known Unscented Kalman Filter (UKF). This involves a simple augmentation of the state to include the noise variables. Subsequently, the process and measurement covariance matrices are included in the covariance matrix of the augmented state.
U.S. 2008/0071469, published Mar. 20, 2008 to Caveney (a co-inventor of the present invention) discloses an alternative approach to using Kalman prediction. This publication describes predicting a future position of an automotive vehicle through NI of a non-linear model. Numerical Integration for Future Vehicle Path Prediction by co-inventor Caveney also describes a method for predicting a future position of an automotive vehicle through NI of a non-linear model. The U.S. 2008/0071469 publication, specifically the NI techniques described therein, is hereby incorporated in its entirety by reference.